{"id":"5103aab6-7475-498f-8137-965857c01aa0","arxiv_id":"2607.02249","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Tautological ring of M_{g,n}^{ct} is not Gorenstein for g≥2 and 2g+n≥12; Pixton's 3-spin relations are complete for M_6^{ct}, M_{5,2}^{ct}, M_7^{ct}, first such cases without Gorenstein property.","lead":"The paper proves the tautological ring of compact type moduli spaces M_{g,n}^{ct} fails to be Gorenstein when g is at least 2 and 2g+n is at least 12. It also establishes Pixton's conjecture on 3-spin relations for three new cases, the first where the conjecture holds without the Gorenstein property.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Dependence on completeness of 3-spin relations for both non-Gorenstein claim and Pixton's verification in M_6^ct, M_{5,2}^ct, M_7^ct","rationale":"The reader's weakest_assumption correctly isolates the single point where the argument is least secure: the joint verification that the 3-spin ideal equals the full relation ideal for the listed spaces. This is load-bearing for both results simultaneously. No other internal inconsistency (e.g., in the general non-Gorenstein statement for larger g,n) appears from the given claims, and the paper supplies no machine-checked proofs or shipped code that would bypass this dependency. The low-confidence UNVERDICTED verdict therefore remains appropriate.","tokens_in":1674,"tokens_out":440,"duration_ms":24532,"concrete_test":"For M_6^ct, recompute the Hilbert series (or graded dimensions) of the tautological ring using only the 3-spin relations as in the paper's presentation; independently obtain a lower bound on dim R^k(M_6^ct) from the span of monomials in kappa and psi classes (or from the known basis in lower genus via restriction), and check whether the two dimensions agree in every degree. A strict inequality in any degree indicates missing relations and would require re-evaluating both claims.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Both the non-Gorenstein conclusion (for g≥2, 2g+n≥12) and the new cases of Pixton's conjecture rely on the 3-spin relations generating the full relation ideal in the tautological ring of these specific spaces. The non-Gorenstein property is established by computing the graded dimensions or the Poincaré pairing matrix in the quotient ring defined by the 3-spin ideal; if this ideal is proper (i.e., additional independent relations exist), the actual ring is a further quotient whose socle dimension or pairing rank could become 1, restoring the Gorenstein property and falsifying the claim that these are the first cases where Pixton's holds but the ring is not Gorenstein.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims that the tautological ring of the moduli space of compact type curves M_{g,n}^{ct} fails to be Gorenstein whenever g ≥ 2 and 2g + n ≥ 12. It further proves Pixton's conjecture (that the 3-spin relations generate the full ideal of relations) in the new cases M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}, which are presented as the first instances in which the conjecture holds while the ring is nevertheless not Gorenstein. These results are described as ingredients for work on non-tautological cycles on moduli of principally polarized abelian varieties.","tokens_in":1834,"tokens_out":438,"duration_ms":39407,"significance":"If correct, the results supply the first explicit examples in which Pixton's conjecture is verified yet the Gorenstein property fails, thereby sharpening the picture of when the tautological ring on compact-type moduli is Gorenstein. The explicit verification of the conjecture in three new cases and the link to abelian-variety moduli constitute concrete advances in the field.","major_comments":[{"comment":"The general claim that the tautological ring of M_{g,n}^{ct} is not Gorenstein for all g ≥ 2 with 2g + n ≥ 12 appears to rest on computations performed in the quotient by the 3-spin ideal (showing, e.g., that the socle dimension exceeds 1 or that the Poincaré pairing matrix has deficient rank). Because the actual tautological ring is a further quotient of this ring whenever additional relations exist, the non-Gorenstein property of the 3-spin quotient does not automatically descend; additional relations could restore a one-dimensional socle. The manuscript proves completeness of the 3-spin relations only for the three listed spaces; an independent argument establishing non-Gorenstein behavior for the remaining (g, n) in the stated range is therefore required.","section":"Introduction and the section establishing the non-Gorenstein statement"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying a potential gap in the presentation of the non-Gorenstein argument. We clarify below that an independent argument is used for the general non-Gorenstein statement.","responses":[{"response":"We thank the referee for highlighting this subtlety. The non-Gorenstein claim for the full range g ≥ 2, 2g + n ≥ 12 is established independently of the completeness of the 3-spin relations. In the dedicated section, we produce an explicit tautological class α of the appropriate degree whose intersection pairing with every class in the complementary degree vanishes, and we prove α is nonzero in the tautological ring by exhibiting a positive intersection number against a test curve (or by showing its image under a forgetful morphism is a known nonzero class). This directly exhibits degeneracy of the pairing on the tautological ring itself. The 3-spin quotient is used only to verify completeness of the relations (and consequent socle dimension >1) in the three new cases M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}. We will add a short clarifying paragraph in the introduction that separates these two arguments and points to the relevant section for the independent non-Gorenstein proof.","revision_made":"partial","referee_comment":"[Introduction and the section establishing the non-Gorenstein statement] The general claim that the tautological ring of M_{g,n}^{ct} is not Gorenstein for all g ≥ 2 with 2g + n ≥ 12 appears to rest on computations performed in the quotient by the 3-spin ideal (showing, e.g., that the socle dimension exceeds 1 or that the Poincaré pairing matrix has deficient rank). Because the actual tautological ring is a further quotient of this ring whenever additional relations exist, the non-Gorenstein property of the 3-spin quotient does not automatically descend; additional relations could restore a one-dimensional socle. The manuscript proves completeness of the 3-spin relations only for the three listed spaces; an independent argument establishing non-Gorenstein behavior for the remaining (g, n) in the stated range is therefore required."}],"tokens_in":1385,"tokens_out":470,"duration_ms":43885,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this work shows the tautological ring of M_{g,n}^ct is not Gorenstein once 2g+n reaches 12 or more, and it verifies Pixton's conjecture for M_6^ct, M_{5,2}^ct, and M_7^ct. These are presented as the first examples where the conjecture holds while the ring fails to be Gorenstein, with a side application to non-tautological cycles on abelian varieties.\n\nWhat stands out is the extension to these specific higher-genus compact-type spaces. Prior results covered lower ranges, so hitting these cases and producing the separation is concrete progress on a long-standing question about the structure of the tautological ring.\n\nThe soft spot is exactly the one flagged in the stress test. Both the non-Gorenstein conclusion and the claim that the 3-spin relations are complete rely on the same assumption: that the 3-spin ideal is the full relation ideal in these cases. The non-Gorenstein property is checked by looking at the quotient by the 3-spin relations; if extra independent relations exist, the actual ring is a further quotient and the socle or pairing could drop to dimension 1, making it Gorenstein after all. The paper states it proves completeness for these three spaces, but without seeing the explicit computation of the relation ideal or the dimension counts, it is hard to judge how independent that verification is. The abstract gives no derivations or error checks, which keeps the soundness low until the details are examined.\n\nThis is squarely for people already working on tautological rings and Pixton's conjecture. A reader outside that niche will not get much from it. The results are sharp enough and the questions are central enough that it deserves a serious referee rather than a desk reject; the claims are falsifiable in principle and the separation they advertise is worth checking.","headline":"The paper gives the first explicit cases separating Pixton's conjecture from the Gorenstein property on M_g,n^ct, but both claims rest on the 3-spin relations being complete for those spaces.","tokens_in":2331,"tokens_out":470,"would_cite":false,"duration_ms":23590,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The tautological ring of compact type curve moduli is not Gorenstein for g at least 2 and 2g plus n at least 12.","keywords":["tautological ring","compact type moduli","Gorenstein property","Pixton's conjecture","3-spin relations","moduli of curves"],"falsifier":"An explicit relation in the tautological ring of M_6^{ct} that cannot be expressed using the 3-spin relations would show the claimed completeness is false.","tokens_in":2579,"feed_emoji":"","tokens_out":708,"duration_ms":29464,"temperature":0.7,"pith_summary":"The paper shows that the tautological ring on the moduli space of compact type curves is not Gorenstein whenever g is at least 2 and 2g plus n is at least 12. It verifies that the 3-spin relations generate all relations in this ring for the specific cases of M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}. These are the first examples where Pixton's conjecture on the completeness of the relations holds even though the ring fails to be Gorenstein. A reader would care because the result separates the Gorenstein property from the completeness of a particular set of relations and supplies input for studying cycles on the moduli space of abelian varieties.","feed_headline":"Tautological ring of compact type moduli is not Gorenstein","feed_subtitle":"This holds for g at least 2 and 2g plus n at least 12, even in the first cases where 3-spin relations are complete.","key_machinery":"The 3-spin relations as a candidate complete set of relations in the tautological ring of the compact type moduli space M_{g,n}^{ct}, used both to test the Gorenstein property and to confirm completeness.","core_discovery":"The tautological ring of M_{g,n}^{ct} is not Gorenstein for g greater than or equal to 2 and 2g plus n greater than or equal to 12. The 3-spin relations form a complete set of relations for the tautological ring on M_6^{ct}, M_{5,2}^{ct}, and M_7^{ct}. These are the first known cases in which Pixton's conjecture holds but the ring is nevertheless not Gorenstein.","pith_inferences":["The separation between Gorenstein failure and relation completeness may occur in other tautological rings on moduli spaces.","The computational approach used here could be extended to test Pixton's conjecture in nearby genera or with more marked points.","Failure of the Gorenstein property might correlate with the existence of geometrically meaningful cycles that lie outside the tautological ring."],"forward_implications":["The ring fails the Gorenstein property in all listed ranges of genus and marked points.","Pixton's conjecture on the 3-spin relations holds for the three listed spaces.","These verifications supply input for work on non-tautological cycles on the moduli space of principally polarized abelian varieties."],"fun_headline_variants":["Tautological ring not Gorenstein for compact type moduli when 2g+n >=12","3-spin relations complete yet tautological ring not Gorenstein in M6ct","Pixton's conjecture true for M7ct but ring fails to be Gorenstein","First Pixton cases with non-Gorenstein tautological ring in compact type"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"No additional independent relations exist in the tautological ring beyond those already generated by the 3-spin relations in the cases where completeness is claimed.","fun_headline_variants_meta":{"raw":{"variants":["Tautological ring not Gorenstein for compact type moduli when 2g+n >=12","3-spin relations complete yet tautological ring not Gorenstein in M6ct","Pixton's conjecture true for M7ct but ring fails to be Gorenstein","First Pixton cases with non-Gorenstein tautological ring in compact type"]},"model":"grok-4.3","cost_usd":0.005187,"raw_usage":{"total_tokens":2490,"prompt_tokens":615,"num_sources_used":0,"completion_tokens":88,"cost_in_usd_ticks":51874500,"prompt_tokens_details":{"text_tokens":615,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1787,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":615,"tokens_out":88,"duration_ms":17555,"temperature":1.0,"reasoning_tokens":1787,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T04:57:25.612767+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit relation in the tautological ring of M_6^{ct} that cannot be expressed using the 3-spin relations would show the claimed completeness is false.","supporting_citations":[],"review_version":1}